Linear Equations in Two Variables

Introduction

Linear equations in two variables are an important part of algebra. They are used to represent relationships between two unknowns and play a crucial role in solving various problems. Such equations are widely applied in fields like mathematics, physics, and economics.

A linear equation in two variables is an equation of the form:

ax + by + c = 0

Where:

• a, b, and c are real numbers

• x and y are variables

• a and b are not both zero
  
  

Each solution of the equation represents a point (x, y) on a straight line when plotted on a graph.

It is important to understand linear equations in two variables since they form the basis for further complex mathematical concepts and real-world applications in mathematics, physics, business, engineering, and economics. Through understanding various forms, such as standard form, slope-intercept form, and intercept form, and solution techniques, such as graphical, substitution, and elimination, studying linear equations in two variables makes it easier to solve problems and understand links between things

 

Table of Contents

• What are Linear Equations in Two Variables?
• Forms of Linear Equations in Two Variables
• Solving Pairs of Linear Equations in Two Variables
• Graphical Method for Solving Linear Equations
• Algebraic Methods of Solving Linear Equations
  • Substitution Method
  • Elimination Method
• Linear Equations in Two Variables Examples
• Linear Equations in Two Variables Questions
• Conclusion
• FAQs on Linear Equations in Two Variables

 

What are Linear Equations in Two Variables?

A linear equation in two variables is an algebraic expression where both variables (x and y) have a degree of 1. These equations always represent straight lines on the Cartesian plane and have infinitely many solutions.

Examples:

• 2x+3y=6

• x−y=5

• 4x+0y=12(also linear in one variable)
  
  

These are common linear equations in two variables examples.

 

Forms of Linear Equations in Two Variables

1. Standard Form:
   Ax+By+C=0
   
   Example:3x+4y−7=0
   
   

2. Slope-Intercept Form:
   y=mx+c
   
   Example: y=2x+5
   
   

3. Intercept Form:
   xa+yb=1
   
   Example:x2+y3=1
   
   

Understanding these forms makes it easier to solve linear equations in two variables.

 

Solving Pairs of Linear Equations in Two Variables

When two linear equations in two variables are given, solving them means finding a pair (x, y) that satisfies both equations. This pair is the point where their graphs intersect.

Example:
x+y=5

2x−y=4

This pair can be solved by the graphical method or algebraic methods like substitution and elimination.

 

Graphical Method for Solving Linear Equations in Two Variables

The graphical method is a visual way to solve a pair of linear equations in two variables. In this method, we draw the graphs (straight lines) of both equations on the same coordinate plane. The point where the two lines intersect gives the solution to the system of equations.

Steps to Solve Linear Equations Graphically:

Step 1: Convert the equations into the form
  If the equations aren't already in slope-intercept form, convert them to y = mx + c.

Step 2: Make a table of values
      Determine the matching y-values for each equation by selecting at least two x-values.

Step 3: Plot the points
  On a graph paper, mark the points for each equation from the table.

Step 4: Draw the lines
  Draw straight lines connecting the plotted points with a ruler.

Step 5: Find the point of intersection
  The solution to the two equations is found at the intersection of the two lines. The values of x and y that satisfy both equations are provided at this point.

 

Example: Solve the following equations graphically

Equation 1: x+y=6
Equation 2: x−y=2

Step 1: Convert to y=mx+c form

From Equation 1: y=6−x
From Equation 2: y=x−2

Step 2: Prepare a table of values

For y=6−x:

 

For y=x−2:

 

Steps 3 and 4: Plot the points and draw the lines

• Plot the points from both tables on graph paper.

• Draw two straight lines using a ruler for each equation.

Step 5: Find the point of intersection

• The two lines intersect at the point (4, 2).

Solution: x = 4, y = 2

This is the solution to the given pair of linear equations in two variables. It satisfies both equations.

 

This method is ideal when you want to visualize how the two variables relate and where they meet.

 

Algebraic Methods of Solving a Pair of Linear Equations

There are two main algebraic methods for solving linear equations in two variables:

Substitution Method

To solve a system of two linear equations in two variables using the substitution method, follow the steps given below:

Steps to Solve Linear Equations Using the Substitution Method:

Step 1: Solve one of the equations for one variable in terms of the other.
  For example, from the equation x + y = 5, we can write:
  x = 5 - y or y = 5 - x

Step 2: Substitute this expression into the other equation.
  This will give you an equation with only one variable.

Step 3: Solve the resulting equation to find the value of the single variable.

Step 4: Substitute the value obtained back into the expression from Step 1 to find the second variable.

Step 5: Write the solution as an ordered pair (x, y).
  Check the solution by plugging the values back into both original equations.

 

Example: Solve the following system using the substitution method

Equation 1: x+y=5
Equation 2: x−y=1

 

Step 1: Solve Equation 1 for x:
  x=5−y

Step 2: Substitute into Equation 2:
  (5−y)−y=1
  5−2y=1

Step 3: Solve for y:
  −2y=1−5
  −2y=−4
  y=2

Step 4: Substitute y = 2 into x = 5 - y:
  x=5−2
  x=3

Solution:
x = 3, y = 2
Ordered pair: (3, 2)

Check:
Equation 1: 3+2=5
Equation 2: 3−2=1 

This is how you solve linear equations in two variables using the substitution method. It is especially useful when one variable is already isolated or can be easily isolated.

 

Elimination Method

The elimination method is used to solve a system of two linear equations in two variables by eliminating one of the variables. This is done by adding or subtracting the equations so that one variable cancels out, making it easier to solve for the other.

 

Steps to Solve Linear Equations Using the Elimination Method:

Step 1: Write both equations in standard form:
  Ax + By = C

Step 2: Make the coefficients of one variable (either x or y) the same in both equations.
  You may need to multiply one or both equations.

Step 3: Add or subtract the equations to eliminate one variable.

Step 4: Solve the resulting single-variable equation.

Step 5: Substitute the value of the solved variable into either original equation to find the second variable.

Step 6: Write the final answer as an ordered pair (x, y) and check your solution.

 

Example: Solve the following system using the elimination method

Equation 1: 3x+2y=16
Equation 2: 2x−2y=4

Step 1: Equations are already in standard form.

Step 2: Add both equations to eliminate y:

(3x+2y)+(2x−2y)=16+4

→5x=20

Step 3: Solve for x:
x=20÷5=4

Step 4: Substitute x = 4 into Equation 1:

3(4)+2y=16
12+2y=16
2y=4
y=2

Solution:
x=4,y=2
Ordered pair: (4, 2)

Check:
Equation 1:3(4)+2(2)=12+4=16
Equation 2: 2(4)−2(2)=8−4=4

 

This is how you solve linear equations in two variables using the elimination method. It’s especially efficient when the coefficients of one variable are already equal or can be easily made equal.

Linear Equations in Two Variables Examples

Example 1: Solve using the substitution method
x+y=9x−y=3

Step 1: From the first equation, solve for x:
x=9−y

Step 2: Substitute into the second equation:
(9−y)−y=39−2y=3−2y=3−9=−6y=3

Step 3: x=9−3=6

Answer: x=6,y=3

 

Example 2: Solve using the elimination method
2x+3y=124x−3y=6

Step 1: Add both equations to eliminate y:
(2x+3y)+(4x−3y)=12+66x=18x=3

Step 2: Substitute into the first equation:
2(3)+3y=126+3y=123y=6y=2

Answer:x=3,y=2

 

Example 3: Solve using the graphical method
x+y=4x−y=0

Convert to slope-intercept form:
Equation 1: y=4−x
Equation 2: y=x

Table of values:

For y = 4 - x:

 

For y = x:

Plot both lines on graph paper → They intersect at (2, 2)

Answer:x = 2, y = 2

 

Example 4: Solve using the substitution method
3x−y=7x+y=5

Step 1: Solve the second equation for y:
y=5−x

Step 2: Substitute in the first equation:
3x−(5−x)=73x−5+x=74x=12x=3

Step 3:y=5−3=2

Answer:x=3,y=2

 

Example 5: Solve using the elimination method
4x+5y=232x−5y=−1

Step 1: Add both equations to eliminate y:
(4x+5y)+(2x−5y)=23+(−1)6x=24x=12x=113

Step 2: Substitute into the second equation:
2(11/3)−5y=−122/3−5y=−1−5y=−1−22/3−5y=(−3−22)/3=−25/3y=5/3

Answer: x=113,y=53

 

Linear Equations in Two Variables Questions

Try solving these linear equations in two variables questions:

1. Solve using the substitution method:
x+y=12
x−y=4

2. Solve using the elimination method:
2x+3y=18
4x−3y=6

3. Solve the following system graphically:
x+y=5
2x−y=1

4. Find the values of x and y:
3x+2y=10
4x−y=7

5. Check whether the system has a unique solution, no solution, or infinitely many solutions:
2x−3y=6
4x−6y=12

6. Solve:
x=2y+3
x−y=7

7. Solve using the elimination method:
5x+4y=20
10x+8y=40

8. Solve the system:
0.5x+y=5
1.5x−2y=3

9. Solve the following:
x+3y=6
2x−y=8

10. Solve using substitution:
x=y−4
3x+2y=16

These help in practicing solving linear equations in two variables.

 

Conclusion

Mastering linear equations in two variables is a key math skill. Whether using the graphical method, substitution, or elimination, it’s essential to practice regularly. Use different linear equations in two variables examples to build confidence and tackle a wide range of problems.

Practice with multiple linear equations in two variables questions to solidify your understanding.

 

Frequently Asked Questions on Linear Equations in Two Variables

1. What is the formula for a linear equation in two variables?

Answer: The standard formula is:
ax + by + c = 0
where x and y are variables, and a, b, and c are real numbers.
Note: a and b cannot both be zero.

 

2. What are 5 examples of linear equations in two variables?

Answer:

1. x + y = 7

2. 2x - 3y = 6

3. 4x + 5y = 20

4. x - 2y + 3 = 0

5. 6x + y = 12

 

3. Is 2x + 3y + 7 linear or nonlinear?

Answer: Linear.Because both x and y are to the power of 1 and there are no products or functions like square roots or exponents.

 

4. Which is the linear equation in two variables: 5x + 3y = 7, 5x + 9 = 0, 6y = 7, none of these?

Answer:5x + 3y = 7
It contains two variables (x and y), both of degree 1.

 

5. Is 5x + 3y = 11 and 10x + 6y = -22 consistent or inconsistent?

Answer: Inconsistent.
Multiplying the first equation by 2 gives:
10x + 6y = 22
This conflicts with the second equation:
10x + 6y = -22
Same left-hand side, different right-hand sides ⇒ No solution ⇒ Inconsistent system.